Black hole formula
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Black Hole Formulas: A Comprehensive Overview
Bekenstein-Hawking Entropy Formula
The Bekenstein-Hawking entropy formula is a cornerstone in black hole physics. It states that the entropy ( S ) of a black hole is proportional to the area ( A ) of its event horizon, given by ( S = \frac{A}{4 \ell_P^2} ), where ( \ell_P ) is the Planck length. This relationship has been confirmed through various approaches, including loop quantum gravity, which uniquely fixes the value of the 'quantum of area' in the theory Ashtekar1997Bardeen1973Hod2008.
Quantum Corrections to Black Hole Entropy
Recent studies have explored quantum corrections to the Bekenstein-Hawking entropy. These corrections often involve higher curvature terms and conformal field theory techniques. For instance, logarithmic corrections to the Cardy formula provide first-order quantum corrections to the Bekenstein-Hawking entropy, which are proportional to the logarithm of the horizon size . Additionally, in N=4 string theories, the exact entropy of two-charge supersymmetric black holes has been computed using Wald's formula, showing agreement between macroscopic and microscopic entropy calculations .
Mass and Energy Transformations in Black Holes
The mass of a black hole can be expressed as a function of its irreducible mass, angular momentum, and charge. For example, the formula ( E^2 = m_{\mathrm{ir}}^2 + \left(\frac{L^2}{4m_{\mathrm{ir}}^2}\right) + p^2 ) describes a black hole with linear momentum ( p ) and angular momentum ( L ) . Furthermore, it has been shown that up to 50% of the mass of an extreme charged black hole can be converted into energy, compared to 29% for an extreme rotating black hole .
Higher Curvature Interactions and Black Hole Entropy
In Lovelock higher-curvature gravity theories, the entropy of stationary black holes is not simply one quarter of the horizon area. Instead, it includes a sum of intrinsic curvature invariants integrated over a cross-section of the horizon. This general formula is derived by integrating the first law of black hole mechanics using Hamiltonian methods .
Relaxation of Rapidly Rotating Black Holes
The relaxation phase of perturbed, rapidly rotating black holes has been studied analytically. A simple formula for the fundamental quasinormal resonances of near-extremal Kerr black holes is given by ( \omega = m\Omega - i2\pi T_{\mathrm{BH}}(n + \frac{1}{2}) ), where ( T_{\mathrm{BH}} ) and ( \Omega ) are the temperature and angular velocity of the black hole, respectively. This formula indicates that the relaxation period becomes extremely long as the extremal limit ( T_{\mathrm{BH}} \rightarrow 0 ) is approached .
Conclusion
The study of black hole formulas encompasses a wide range of topics, from the foundational Bekenstein-Hawking entropy to quantum corrections and the dynamics of rotating black holes. These formulas not only deepen our understanding of black hole thermodynamics but also bridge the gap between classical and quantum theories of gravity.
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